Brauer-Severi varieties associated to twists of the Burkhardt quartic

The Burkhardt quartic is a projective threefold which, geometrically, is birational to the moduli space of abelian surfaces with full level-3 structure. We study this moduli interpretation of the Burkhardt quartic in an arithmetic setting, over a general field $k$. As it turns out, some twists of the Burkhardt quartic have a nontrivial field-of-definition versus field-of moduli obstruction. Classically, if a twist has a $k$-rational point then the obstruction can be computed as the Brauer class of an associated conic. Using representation theory, we show how to compute the obstruction without assuming the existence of a $k$-rational point, giving rise to an associated 3-dimensional Brauer-Severi variety rather than a conic. This Brauer-Severi variety itself has a related moduli interpretation.